Impedance Matching and Matching
Networks
Valentin Todorow,
December, 2009
ETCH PRODUCTS BUSINESS GROUP
Valentin Todorow 2009
RF for Plasma Processing - Definition of RF
What is RF?
The IEEE Standard Dictionary of Electrical and Electronics Terms
defines RF as:
Radio frequency is the frequency in the portion of the
electromagnetic spectrum that is between the audio frequency
and infrared portion. Typically between 20 kHz and 300 GHz.
Band designation:
–
–
–
–
HF
VHF
UHF
Microwave
3 – 30 MHz
30 – 300 MHz
300 – 1000 MHz
above 1 GHz
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Valentin Todorow 2009
RF for Plasma Processing
Why RF for Plasma Processing?
1. To sustain plasma a closed loop circuit is needed. Current need to flow
from generator through the wafer, plasma, chamber wall back to the
generator.
DC would be perfect for PVD sputtering because the target is
conductive and DC current can flow through it. In the case of a
semiconductor wafer which is under normal conditions dielectric DC
current can not flow through it and plasma can not be sustained. From
basic electrical science we know that AC current can flow through
dielectrics. So an AC energy is needed drive current through a dielectric
wafer for plasma processing.
2. Plasma properties: Different frequencies create different plasmas
densities and sheath voltages.
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Impedance Matching – Why?
Impedance matching is needed to provide maximum power transfer
between the source or RF energy and its load. This is especially
important if you deal with low amplitude signals. Imagine a radio or
TV antenna. To get a good reception every bit of this signal needs to
be used and the designer can not afford any signal loss – a perfect
match is desired. So the first reason for matching is just power
efficiency.
The second reason is device protection – If RF circuit is not matched
we get reflected power. This reflected power builds standing waves
on the transmission line between the source and load. Depending on
the phase between the forward and reflected both waves can either
subtract or add. Because of that on the line we can get places where
the voltage is the sum of both voltages or eventually places where
the voltage equals zero (maximum current). If the standing wave is
positioned in such a way on the transmission line so that the
maximum voltage or current is applied to the power FET’s they can
be destroyed.
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Impedance matching – Definition
What do we mean by impedance matching?
– For DC it is well known theorem that maximum power transfer can be
achieved if source resistance is equal to the load resistance.
– For RF we consider impedances. The condition for impedance matching
is that real part of the impedance should be equal to the real part of the
load and reactance's should be equal and opposite in character. For
example if our source impedance is R + jX to achieve matching our load
should be R – jX.
If we assume that we have a chamber with capacitive discharge the
impedance in general will be R – jX. Generator typical output impedance
is 50 Ohms. Then the matching network has to make 50 = Rl and jX = 0.
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Impedance Matching
Let us match a 50 ohm generator to a 2 Ohm load. One of the ways
to match is to put a resistor in parallel to the output of the generator.
50 Ohm
2.09 Ohms
2 OHm
13.56 MHz
Z=
R1xR2
R1+R2
= 2 Ohm
Matching will be achieved, however not a desirable solution because most
of the power will be lost in the resistor we added. I have chosen here to make
50 Ohm side look as 2 Ohm because it is easier to correlate to the L circuit
which we will derive later.
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Impedance Matching
Now lets see what will happen if we replace the resistor with a
capacitor of 1150 pF which has impedance X=1/2ΠfC =-j10.207
Ohm @ 13.56 MHz
50 Ohm
10.207 Ohms
2 OHm
13.56 MHz
R.(-jX)
Z=
R+(-jX)
= 2 – j 9.8 Ohms
One can see that by adding the capacitor we were able to transform 50 ohm
in to a complex impedance with a real part of 2 Ohms.
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Impedance Matching
One can see that by adding the capacitor we were able to transform
50 ohm in to a complex impedance with a real part of 2 Ohms. So
half of the job is done.
-j9.8
50 Ohm
10.207 Ohms
13.56 MHz
R.(-jX)
Z=
R+(-jX)
2 Ohm
13.56 MHz
= 2 – j 9.8 Ohms
If we find a way to make the imaginary part equal to zero then the match
will be complete.
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Impedance Matching
From electrical science we know that inductors and capacitors have
impedances with opposite signs. Inductors have positive sign and
capacitors negative.
9.8 Ohm
50 Ohm
13.56 MHz
10.207 Ohm
2 Ohm
13.56 MHz
So if we add an inductor which impedance value @ 13.56 MHz is 9.8 Ohm
Inductor and capacitance will cancel each other and our match will be complete.
L=115 nH @13.56 MHz
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Basic L Matching Network
This way we arrived to the basic and most used L shape matching
network. By using capacitors and inductances we can achieve
impedance matching without power loss assuming the components
are ideal. Real capacitors and inductors exhibit losses which need to
be minimized during the match design. The most critical component
is the inductor. At high frequencies the skin effect and inter winding
capacitance decrease the quality of the coil.
X1
X1 :=
Rs
X2
2
( Rs⋅Rp) − Rp
Rp
X2 := Rs⋅
Rp
X1
Rs>Rp
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Basic L Matching Network
Two types L matching networks – Until now we matched real load to
real load. What to do when we have reactance?
High - Low
Low - High
Parallel, also known as shunt or load capacitor is always on the high R side.
Called “load” capacitor because it adjusts the real part of the load. Series
sometimes called “tune” because it tunes out the reactive part of the load.
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RF for Plasma Processing – Matching Networks
There are three basic matching networks used in RF designs - they are L, Π
and T configuration. Each of them has advantages and disadvantages.
Most commonly used matching network in plasma processing is the L circuit.
The reasons are:
Simplicity - Easy to build auto matching networks. It has only two
components to be controlled for adjusting the real and imaginary part of the
impedance.
Practical component values - some of configurations require either
very low inductance values or very high capacitance values which is
impossible to make especially when you have to design auto matching
network with a wide tuning range.
The quality factor Q of the circuit is determined only from the source
(generator) and load (plasma impedance) and does not depend on the
external components - this is may be the most important property of the
circuit. What this means is that for certain load impedance there is only one
combination of L and C to match that load.
Q :=
 Rsource − 1


 Rload 
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Basic L Matching Networks
T and Pi network require more complex control algorithm for
automatic matching network design. Need to control three
components which makes it more expensive.
The disadvantage of the L circuit - it can match loads equal or less
than 50 Ohm. If the L circuit is reversed it can match loads equal or
higher than 50 Ohm. It can not match on both sides. For example If
the load is changing from 35 to 100 Ohms a reversed L network will
match only from 50 up to 100 Ohms and will not match from 35 to 50
Ohm.
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Pi Network
X2
Rs
Q :=
X3
X1
Rl
 Rs 
 
 Rl 
X1 := −Rl⋅
Q2 + 1 −  Rs 

 

 Rl 
Rs>Rl
X21
 Rs  − 1
 
 Rx 
X22
( Q ⋅Rs) −  Rl⋅ Rs 



X1 


X2 :=
2
Q +1
X3
Rx
X1
X3 :=
−Rs
Rx < Rl < Rs
Q
Can be represented as two L circuits connected together matching in to an
Imaginary load Rx which has to be smaller than Rs and Rl and is used to calculate
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Q of the circuit.
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T Network
X3
X1
Q :=
 Rx  − 1
 
 Rl 
Rs
X2
Rl
X3 := Rl⋅Q
X2 := −Rl⋅
Rx > Rs > Rl
( 1 + Q2)
 ( 1 + Q2)  
 Rl⋅
 − 1

Q +


Rs


 ( 1 + Q2) 
−1
X1 := Rs⋅  Rl⋅
Rs 

Could be represented as two L circuits connected together through their shunt
Component. The imaginary resistor Rx this time has to be larger than both Rs
and Rl. Q is defined as the ratio between Rx and the smallest load resistance.
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Smith Chart
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Using Smith Chart to Design Matching Networks
Lets match a Z=2-j35 to 50 ohm which is representative of capacitive plasmas.
We will be using an L circuit and will be moving from the load to the generator. The
first step will be to add the series inductor. The value of the inductor should be
increased
untillGROUP
the trace crosses the 50 Ohm circle.
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Using Smith Chart to Design Matching Networks
The next step will be to add the shunt capacitor. The value is increased until the
trace crosses the 50 Ohm point. The design is ready. Component values can be
red from the chart.
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Using Smith Chart to Design Matching Networks
Lets match a Z=2+j35, which is representative of inductive plasmas. Will be also using
L circuit and will be moving from the load thoward the generator. Normally a L circuit
consists of series inductor. Here the inductor is part of the load and it is larger than
what we need. Because of that our series component will be a capacitor to compensate
for the excessive inductance. Cap Value will decrease untill we cross 50 Ohm circle. 19
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Using Smith Chart to Design Matching Networks
Second step will be to add the shunt component and the impedance transformation
is completed.
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Matching Networks Design Considerations
Match range – determined by plasma impedance range. Usually
determined empirically. One can use existing matches or design its
own box with variable components. Use this matches to manually
tune in to plasma condition. Plasma impedance can be determined
two ways:
– Load the input of your box with a 50 ohm load and measure the
impedance on the match output.
– Measure the values of the series and shunt component and calculate the
impedance.
Why chamber impedance is important?
– Determine range of values for the series and shunt component.
– Determine current and voltage capabilities of the components
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Matching Networks Design Considerations
Determining RF current through the match – We will use the already known
equation for the quality factor:
Q :=
 Rsource − 1


Rload


Use the lowest value for Rload you have measured and calculate Q. From resonant
circuit theory we also know that the current within the circuit = outside current x Q.
outside current can be calculated from Ohms law using the maximum power which
the match will handle and 50 Ohms as source impedance. For example for 3 kW
Current will be:
R := 50
P := 3000
I :=
P
R
I = 7.746
For Rload= 1 ohm Q will be 7. Then match current will be 54A.
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Matching Networks Design Considerations
Series leg design – If you have a look at the L matching network topography
you will see that series leg is an inductor. In auto matches we use variable
components and it is hard to design variable inductors. Because of that a
series combination of L and a variable C is used. From the equation below one
can see that if we vary Xc and Xc<Xl the impedance of the series leg remains
inductive. This way we can have a variable inductor.
X := Xl − Xc
Inductor design consideration – coil is the major loss component in the match.
Because of the high current and skin effect real losses in the coil can be
significant if it is not sized properly. Coil temperatures can reach more that 150
degree C if the cross section is not selected taking in to consideration the skin
dept at the frequency of operation,
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Matching Networks Design Considerations
Shunt leg design – In many cases a large value for the shunt capacitor is
required. In this case the designer has two options
– Use fixed capacitance in parallel with the variable cap. The back draw of such
design is that it shifts the impedance. It expands the range at higher end, however it
limits you at the lower end.
– A better solution is a series combination of variable capacitor C and a small
inductor. When Xc is at its max value and Xl<<Xc the total X is almost unchanged.
So there is no limiting at the upper end of the tuning range. If Xc is at its lower end
and Xc is comparable but higher to Xl the value of X will be drastically changed. For
example if Xc is 2 Ohms and Xl=1 Ohm, the total impedance will be 1 ohm which is
like doubling the capacitance in value. With such design one needs to make sure
that you are not operating at series resonance (Xl=Xc) at the operating frequency
because currents in the leg will be enormous.
X := Xl − Xc
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Matching Networks Design Considerations
Parasitic reactance – every piece of wire will exhibit inductance and
capacitance to ground. Previous circuits we analyzed do not include
any of this stray inductance or capacitance. In real world design all
this should be considered, especially at high frequencies. The rule is
keep wires as short as possible and as far as possible from ground.
If not possible every parasitic should be analyzed and included in
your schematic.
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Matching Networks Design Considerations- Grounding
Straight wire inductor
–
–
–
–
–
–
–
–
–
L = 0.002l[2.3 log(4l/d – 0.75)] uH
l = wire length in cm
d= wire diameter in cm
Lets assume a 2mm diameter wire with 30 cm and 100 cm length
L30 = 0.38 uH
L100 = 1.5 uH
@50 Hz impedance will be:
X30 = 0.00012 Ohm
X100 = 0.00047
@13.56 MHz the impedance will be:
X30 = 32 Ohm
X100 = 128 Ohms
From the above example we can see that what is considered ground
for AC is not a ground for RF. Higher the frequency higher the
impedance of a wire.
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